Abstract
A Carnot group \(\mathbb {G}\) is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. Intrinsic regular surfaces in Carnot groups play the same role as \(\mathbb {C}^{1}\) surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize the results on Ambrosio et al. (J. Geom. Anal., 16, 187–232, 2006) valid in Heisenberg groups to the more general setting of Carnot groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosio, L., Kirchheim, B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318(3), 527–555 (2000)
Ambrosio, L, Serra Cassano, F., Vittone, D.: Intrinsic regular hypersurfaces in heisenberg groups. J. Geom. Anal. 16, 187–232 (2006)
Arena, G., Serapioni, R.: Intrinsic regular submanifolds in heisenberg groups are differentiable graphs. Calc. Var. Partial Differential Equations 35(4), 517–536 (2009)
Bigolin, F., Caravenna, L., Serra Cassano, F.: Intrinsic Lipschitz Graphs in Heisenberg Groups and Continuous Solutions of a Balance Equation. Ann. I. H. Poincaré -AN (2014)
Bigolin, F, Serra Cassano, F.: Intrinsic regular graphs in heisenberg groups vs. weak solutions of non linear first-order PDEs. Adv. Calc. Var. 3, 69–97 (2010)
Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics. Springer-Verlag, Berlin Heidelberg New York (2007)
Di Donato, D.: Intrinsic Differentiability and Intrinsic Regular Surfaces in Carnot Groups. University of Trento. PhD Thesis (2017)
Evans, L.C., Gariepy, R. F.: Measure Theory and Fine Properties of Functions. CRC Press, BocaRaton (1992)
Federer, H.: Geometric Measure Theory. Springer (1969)
Franchi, B, Marchi, M., Serapioni, R.: Differentiability and approximate differentiability for intrinsic lipschitz functions in carnot groups and a rademarcher theorem. Anal. Geom. Metr. Spaces. 2, 258–281 (2014)
Franchi, B., Serapioni, R.: Intrinsic Lipschitz Graphs within Carnot groups. The Journal of Geometric Analysis 26(3), 1946–1994 (2016)
Franchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the heisenberg group. Math. Ann. 321, 479–531 (2001)
Franchi, B., Serapioni, R., Serra Cassano, F.: Regular hypersurfaces, intrinsic perimeter and implicit function theorem in carnot groups. Comm. Anal. Geom. 11(5), 909–944 (2003)
Franchi, B., Serapioni, R., Serra Cassano, F.: Regular Submanifolds, Graphs and area formula in Heisenberg Groups. Advances in Math. 211(1), 152–203 (2007)
Franchi, B., Serapioni, R., Serra Cassano, F.: On the structure of finite perimeter sets in step 2 carnot groups. The Journal of Geometric Analysis 13(3), 421–466 (2003)
Garofalo, N., Nhieu, D.M.: Isoperimetric and sobolev inequalities for carnot-carathéodory spaces and the existence of minimal surfaces. Comm. Pure Appl. Math. 49, 1081–1144 (1996)
Kirchheim, B., Serra Cassano, F.: Rectifiability and parametrization of intrinsic regular surfaces in the heisenberg group. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) III, 871–896 (2004)
Kozhevnikov, A.: Propriétés métriques des ensembles de niveau des applications différentiables sur les groupes de Carnot. PhD Thesis. Université Paris-Sud (2015)
Le Donne, E.: A primer on carnot groups: Homogenous Groups, carnot-carathéodory Spaces, and Regularity of Their Isometries. Anal. Geom. Metr. Spaces 5, 116–137 (2017)
Lu, G.: Embedding theorems into lipschitz and BMO spaces and applications to quasilinear sub elliptic differential equations. Publ. Mat. 40, 301–329 (1996)
Magnani, V: Elements Of Geometric Measure Theory on Sub-Riemannian Groups. PhD Thesis, Scuola Normale Superiore, Edizioni della Normale Pisa (2002)
Magnani, V.: Towards differential calculus in stratified groups. J. Aust. Math. Soc. 95(1), 76–128 (2013)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)
Mitchell, J.: On carnot-carathéodory metrics. J. Differ. Geom. 21, 35–45 (1985)
Pansu, P.: Métriques de carnot-carathéodory et quasiisométries des espaces symétriques de rang un. Ann. of Math. 129, 1–60 (1989)
Serapioni, R.: Intrinsic difference quotients, Harmonic Analysis, Partial Differential Equations and Applications. Birkhäuser (2017)
Serra Cassano, F.: Some topics of geometric measure theory in carnot groups, geometry, analysis and dynamics on sub-riemannian manifolds. European Mathematical Society I, 1–121 (2016)
Vittone, D: Submanifolds in carnot groups. PhD Thesis, Scuola Normale Superiore, Edizioni della Normale, Pisa (2008)
Acknowledgments
We wish to express our gratitude to R.Serapioni and F.Serra Cassano, for having signaled us this problem and for many invaluable discussions during our PhD at University of Trento. We thank A.Pinamonti for important suggestions on the subject. We also thank B.Franchi e D.Vittone for a careful reading of our PhD thesis and of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
D.D.D. is supported by University of Trento, Italy.
Rights and permissions
About this article
Cite this article
Di Donato, D. Intrinsic Differentiability and Intrinsic Regular Surfaces in Carnot Groups. Potential Anal 54, 1–39 (2021). https://doi.org/10.1007/s11118-019-09817-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-019-09817-4